Book: Arithmetica Universalis

Overview
Arithmetica Universalis (1707) is Newton’s systematic account of algebra as he taught it in his early Lucasian lectures, edited and published in Latin by William Whiston without Newton’s authorization. It assembles the principles, techniques, and problem types that defined seventeenth‑century algebra, aiming to render general methods for forming, transforming, and solving equations, and for translating geometric and numerical problems into algebraic form. The work deliberately avoids fluxions and calculus; its emphasis is on symbolic manipulation, structural properties of equations, and rigorous procedures that yield exact or approximate solutions.

Foundations and Notation
Newton opens with the arithmetic of magnitudes and numbers, treating operations with rational and irrational quantities, proportions, powers, and the extraction of roots. He systematizes notation to streamline calculation and generalization, encouraging a compact symbolic language in which identical procedures apply across classes of problems. His presentation of the binomial expansion undergirds many techniques for handling fractional and irrational exponents, enabling controlled approximations and algebraic extractions that link arithmetic practice to broader theoretical claims.

Theory of Equations
A central achievement is the sustained analysis of polynomial equations. Newton details transformations that reduce equations to simpler or “depressed” forms by shifts of the variable, simplifying the structure of cubic and higher equations. He relates roots to coefficients through identities now associated with his name: power sums of roots are connected recursively to the elementary symmetric functions appearing as coefficients. This framework extends Viète’s relations and supplies a powerful device for deducing information about roots without explicit solution. He also develops procedures for locating and bounding roots, adapted from and refining the spirit of Descartes’ rule of signs, with attention to sign changes, synthetic divisions, and systematic trial that constrain the number and intervals of positive and negative roots.

Approximation and Extraction
Alongside exact relations, Newton promotes practical methods for numerical approximation. Iterative schemes for extracting roots and solving equations are presented as orderly corrections that converge rapidly under suitable conditions, dovetailing with series expansions and the binomial theorem. The interplay between algebraic transformation and numerical refinement illustrates his view that analysis proceeds by reducing complexity and then honing accuracy, rather than by ad hoc tricks tailored to special cases.

Indeterminate Analysis and Number
The treatise addresses indeterminate (Diophantine) problems, seeking rational or integer solutions when equations have more unknowns than constraints. Newton exhibits parameterizations and reductions that convert families of problems, such as generating rational right triangles or finding numbers with prescribed relations, into solvable algebraic forms. He favors methods of descent and transformation that preserve rationality, reinforcing a pragmatic algebraic number theory oriented toward constructive solution rather than abstract existence theorems.

Algebra and Geometry
Geometry serves as a proving ground for algebraic method. By assigning magnitudes to variables and setting up proportions, Newton shows how loci, constructions, and area or ratio problems can be reduced to equations and treated symbolically. While he stops short of the full coordinate geometry and curve classification that appear in his other works, the text insists that geometric complexity often dissolves under appropriate algebraic representation, and that construction rules emerge from the structure of the resulting equations.

Conceptual Stance and Legacy
Newton treats negative and imaginary quantities with calculated reserve: they function coherently in algebraic relations and bookkeeping of roots, even if their “reality” is viewed cautiously. The book’s style is spare and procedural, building competence through general rules and emblematic problems rather than speculative discussion. As an algebraic summa, Arithmetica Universalis shaped instruction in Britain for generations, influencing de Moivre, Maclaurin, and Euler. Its enduring contributions, especially the identities linking coefficients and power sums, disciplined equation reduction, and systematic approximation, helped fix the agenda of eighteenth‑century algebra and provided a durable toolkit for both pure and applied analysis.